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# Recurdyn Solver - Theoretical Manual

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=33 32 3123 22 2113 12 11a a aa a aa a a| (2-1) where , , and h are unit vectors along the f g x , y , and axes, respectively. The zz y x frame is the body reference frame and the frame is the inertial reference frame. Z Y X ZXYrprpxyzso Fig. 2.1 Coordinate systems and a rigid body Velocities and virtual displacements of point in the X O Z Y frame are defined as (2-2a) ((

wr&((

r (2-2b) Their corresponding quantities in the z y x frame are defined as RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION (2-3a) ((

((

=w Ar AwrYTT& & (2-3b) ((

((

= TTAr A r 1.2.2 RELATIVE KINEMATICS FOR A PAIR OF CONTIGUOUS BODIES A pair of contiguous bodies is shown in Fig. 2.2. Body 1) (i is assumed to be an inboard body of body and the position of point is iiO 1) i(i 1)i (i 1)i (i 1) (i i + + = s d s r r (2-4) The angular velocity of body in its local reference frame, using Eq. 2-3a and defining , is iiT1) (i 1)i (iA A A = 1)i (i 1)i (iT1)i (i 1) - (iT1)i (i i + = q H A w A w & (2-5) where is determined by the axis of rotation. H ziXZYri-1ris(i-1)isi(i-1)yi-1xi-1zi-1zi-1xi-1xiyixiziyiyi-1d(i-1)ioi oi-1 Fig. 2.2 Kinematic relationship between two adjacent rigid bodies 1-5 Differentiation of Eq. 2-4, using Eq. 2-3a, yields 1)i (i'1)i (i 1) (i'i'1) i(i i'1) (i'1)i (i 1) (i'1) (i'1)i (i 1) (i'1) (i 1) (i'i i1)i (i) (~~~ + + =q d As A d As A r A r Aq && & (2-6) where symbols with tildes denote skew symmetric matrices comprised of their vector elements that implement the vector product operation (Ref. 1) and denotes the relative coordinate vector. Substituting of Eq. 2-5 and multiplying both sides of Eq. 2-6 by yields 1)i (iq'iTiA 1)i (i 1)i (i1)i (iT1) i(i 1)i (i 1)i (i1)i (iT1) (i1)i (iT1) - i(i 1)i (i 1)i (i 1)i (i1)i (iT1) (i1)i (iTi)~) ( ()~~~(1)i (i + + + = q H A s A d AA s A d s Ar A rq && & (2-7) where i i i~ = A A& is used. Combining Eqs. 2-5 and 2-7 yields the recursive velocity equation for a pair of contiguous bodies. 1)i (i 1)i2 (i 1) (i 1)i1 (i i + = q B Y B Y & (2-8) where ((

+ ((

= I 0A s A d s IA 00 AB)~~~( 1)i (iT1) i(i 1)i (i 1)i (i 1)i (i1)i (iT1)i (iT1)i1 (i (((

+ ((

= 1)i (i1)i (i1)i (iT1) i(i 1)i (i 1)i (i1)i (iT1)i (iT1)i2 (i~) (1)i (iHH A s A dA 00 ABq (2-9) It is important to note that matrices and are functions of only relative coordinates of the joint between bodies 1)i1 (iB1)i2 (iB(i 1) and . As a consequence, further differentiation of the matrices and in Eq. i1)i2 1)i1 (iB(iB RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION 2-9 with respect to other than yields zero. This property plays a key role in simplifying recursive formulas in Section 7. 1)i (iq1)i1 (i 2Y1)m1 Y1) - (mSimilarly, the recursive virtual displacement relationship is obtained as follows. (2-10) 1)i (i 1)i2 (i 1) (i i + = q B Z B Z 1.3. GENERALIZATION OF THE VELOCITY RECURSIVE FORMULA 1n-1n0 Fig. 3.1 A serial chain mechanism Before proceeding to generalize the recursive velocity formula, the computational equivalence between the recursive method and the velocity transformation method is demonstrated using the mechanical system shown in Fig. 3.1. The Cartesian velocity is obtained by replacing by in Eq. 2-8. mi m 1)m (m 1)m2 (m 1) (m (m m + = q B B Y & (3-1) Substitutions of Eq. 2.8 for , , . . . , and yield Y2) - (mY0Y 1-7 1)m - (m 1)m2 - (m1 - m1 j1)j - (j 1)j2 - (jj - m1 k1)1 - m m)(-k k (m1 k0 1)1 - m m)(-k k ( mq Bq B BY B&&+ )`||.|

\|+= = = + + = + + Y (3-2) Thus, the Cartesian velocity for all bodies is obtained as Y q B Y & = (3-3) where is the collection of coefficients of and B1)i (iq& | |T1 ncT T2T1T0 =nY , , Y , Y , Y Y K (3-4) | |T1 nrT) 1 (T12T01T0 =n nq , , q , q , Y q & K & & & (3-5) where and nc nr denote the numbers of the Cartesian and relative coordinates, respectively. The Cartesian velocity , with a given , can be evaluated either by using Eq. 3-3 or by using Eq. 2-8 with recursive numerical substitution of . Since both formulas give an identical result, and recursive numerical substitution is proven to be more efficient in Ref. 4, matrix multiplication with a given will be evaluated by using Eq. 2-8. ncR YnrR q&iYq B&q&Since in Eq. 3-3 is an arbitrary vector in , Eqs. 2-8 and 3-3, which are computationally equivalent, are actually valid for any vector such that q&nrRnrR x x B X & = (3-6) and 1)i - (i 1)i2 - (i 1) - (i 1)i1 - (i ix B X B X + = (3-7) where is the resulting vector of multiplication of and . As a ncR X B x RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION result, transformation of into is calculated by recursively applying Eq. 3.7 to achieve computational efficiency. nrR xW ncR BxGncR QQ Z* T Q q =+*) ! i(iQ( { + +1 i 1)2Q(Tq ===1 - n0 i = +1 - n0 iTi(iqT=0 1.4. GENERALIZATION OF THE FORCE RECURSIVE FORMULA It is often necessary to transform a vector in into a new vector in . Such a transformation can be found in generalized force computation in the joint space with a known force in the Cartesian space. The virtual work done by a Cartesian force is ncRG B gT=nrR = (4-1) where must be kinematically admissible for all joints in Fig. 3.1. Substitution of Z q B Z = into Eq. 4-1 yields T Q B W (4-2) where . Equation 4-2 can be written in a summation form as Q B QT * +T1) i(i q W (4-3) On the other hand, the symbolic substitution of the recursive virtual displacement relationship Eq. 2-10 into Eq. 4-1, along the chain in Fig. 3.1 starting from the body n toward inboard bodies, yields )})+ +=1 iTi(i 1) S B W (4-4) where 2 i 2 i2)1 1)(i (i1 i0+ ++ ++ + S Q B SS (4-5) 1-9 Equating the right sides of Eqs. 4-3 and 4-4, the following recursive formula for is obtained: * Q ( ) 0 ...., 1, - n i ,1 i 1 i)2 1 i(iT1) i(i*= + + ++ + S Q B Q (4-6) where is defined in Eq. 4-5. 1 i+SSince is an arbitrary vector in , Eqs. 4-5 and 4-6 are valid for any vector in . As a result, the matrix multiplication of is evaluated to achieve computational efficiency by QGncRncR G BT ( )( ) 0 ...., 1, i1 i 1 i)1 1 i(iTin1 i 1 iT1)2 i(i 1) i(i= + = + =+ +++ + + +S Q B S0 SS G B g (4-7) where is the result of . g G BT 1.5. GRAPH REPRESENTATIONS OF MECHANICAL SYSTEMS In the previous section, a serial chain mechanism is considered to derive recursive formulas for and B where is a vector in and G in . In general, a mechanical system may have various topological structures. To cope with the various topological structures, an automatic preprocessing is required for a general purposed program, which employs a relative coordinate formulation. The preprocessing identifies the topological structure of a constrained mechanical system to achieve computational efficiency. A graph theory was used to represent bodies and joints for mechanical systems in Refs. 1 and 4. A node and an edge in a graph represented a body and a joint, respectively. The preprocessing based on the graph theory yielded the path and distance matrices that are provided to automatically decide execution sequences Bx GTxnrRncR RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION for a general purposed program. As an example, a governor mechanism and its graph representation are shown in Figs. 5.1 and 5.2. 4328 7651U1R2 R3U2S1 S2T2R1T1: Cut joint Fig. 5.1 Governor mechanism 76453821R1T1R3U2R2U1T2Cut JointCut Joint Fig. 5.2 Graph representation of the governor mechanism 1.6. EQUATIONS OF MOTION AND DAE SOLUTION METHOD The variational form of the Newton-Euler equations of motion for a constrained mechanism is 1-11 0 ) Q Y M Z T= +&( (6-1) where Z must be kinematically admissible for all joints except cut joints [1]. In the equation, and , respectively, denote the cut joint constraint and the corresponding Lagrange multiplier. The mass matrix and the force vector are defined as MQ (nbd 2 1, , , diag M M M M L = ) (6-2) ((

=iiiJ 00 I mM (6-3) | |TnbdT3T2T1, , , , Q Q Q Q Q L = (6-4) ~((

= J nr m fQ~i& (6-5) where denotes the number of bodies, denotes the identity matrix, denotes the moment of inertia, denotes the external force, and denotes the external torque. Substituting the virtual displacement relationship into Eq. 6-1 yields nbd I Jf n { } 0 ) Q Y M B q T= +&(T (6-6) Since q is arbitrary, the following equations of motion are obtained: 0 ) Q Y M B F T= + = &( (6-7) The equations of motion, the constraint equations, q v = & , and constitute the following differential algebraic equations[8]: a v = & RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION 0a vv qa v q v q q ) a , v q ( F=(((((((((

) ( ) ( ) (&&& &&t , , ,t , ,t ,t , , , (6-8) Application of 'tangent space method' in Ref. 7 to Eq. 6-8 yields the following nonlinear system that must be solved at each time step: 0 a v U v q Ua v q v q q ) a , v q ( Fp H =(((((((((

+ + + + ) ( ) ( ) (=) () (t , , ,t , ,t ,t , , ,) (2 n 0 nT01 n 0 nT0n n nn n nn nn n n n nnn& && (6-9) where | |T TnTnTnTn n , a , v , q p =0U, , , and are determined by the coefficients of the BDF, and is an 01nr2ncut) - (nr such that the augmented square matrix is nonsingular. (((

qUT0Applying the Newton's method to solve the nonlinear system in Eq. 6-9 yields H p Hp = (6-10) 1,2,3,... i ,in1 in = + =+p p p (6-11) where 1-13 (((((((((

=0 U U 00 0 U U0 0 0 0 0 0 F F F FHT0a v qv qqq q q qpT0 0T0 0T0 & & & & & && & (6-12) Since and are highly nonlinear functions of , , , and F q v a , care must be taken in deriving the non-zero expressions in , so that they can be efficiently evaluated. pH 1.7. GENERALIZED RECURSIVE FORMULAS Inspection of the residual and Jacobian matrix shows that types of necessary recursive formulas are classified into Bx , , , , HpHG BTx B& ( )qBx( )qG BT, ( )qx B&, and ( )vx B&, where into G are arbitrary constant vectors, and are relative coordinates. Formulas and were derived in Sections 3 and 4, and the formulas for the rest will be derived in this section. All recursive formulas are tabulated in Appendix A. Note that the recursive formulas are quite simple. This simplicity is achieved by exploiting the relative kinematics in the local reference frame instead of the global reference frame. nrR xncRBxq G BTTo derive the formulas systematically, bodies in a graph are divided into four disjoint sets (associated with a generalized coordinate ) as follows: kq( )kq q I having joint the of body outboard body outborad {k =} coordinate d generalize its as ( ) ( ) { }k k of bodies outboard all q I q II = ( ) ( ))`=body inboard and base the including , of body inboard the and body base e between th bodies allkkq Iq III RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION ( ) ( ) ( ) ( ) { }k k k k of set ary complement the q III q II q I q IV U U = For example, the body sets associated with q (relative coordinate between bodies 2 and 4) for the graph shown in Fig. 5.2 are obtained as follows: 24 ( ) { 4 Body 24 = q I }}}} ( ) { 7 and 6 Bodies24 = q II ( ) { 2 and 1 Bodies24 = q III ( ) { 8 and 5 3, Bodies24 = q IV 1.7.1 RECURSIVE FORMULA FOR x B X & & = Recursive formula for is easily obtained by differentiating Eq. 3-7. ncR x B& 1)i (i 1)i2 (i 1 i 1)i1 (i 1 i 1)i1 (i i + + = x B X B X B X & & & & (7-1) This recursive formula can be applied to compute the Cartesian acceleration with known relative velocity and acceleration. Y& 1.7.2 RECURSIVE FORMULA FOR RM BOLD q qBx X ) ( = To obtain the recursive formula for , Eq. 3-7 is partially differentiated with respect to . qBx) (nr ..., 1, k ,k = q 1)i (i 1)2 (i 1 - i 1)i1 - (i 1 i 1)i1 (i ik k k) ( ) ( ) ( ) ( + + = x B X B X B Xq q q qk (7-2) Since matrices and depend only on the relative coordinates for joint , their partial derivatives with respect to generalized coordinates other than are zero. In other words, the partial derivatives are zero if 1)i1 (iB1)i11)i2 (iB1)i - (iq(i kq 1-15 does not belong to set . Therefore if body i is an element of set , Eq. 7-2 becomes ( )kq I II( )kq(q III( )kqik) = Xq24TG) (B k k) ( ) (1 i 1)i1 (i i q qX B X = (7-3) If body i belongs to set ) ( )k kq IV U , is not affected by . As a result, Eq. 7-3 is further simplified as follows iXkq 0 Xq =k) (i (7-4) If body i is an element of set , body is naturally its inboard body and it belongs to set . Using Eq. 7-4, Eq. 7-2 becomes (kq I ) 1) - (iIII 1)i (i 1)2 (i 1 i 1)i1 (ik k) ( ) ( ( + x B X Bq q (7-5) This recursive formula can be applied to compute the partial derivative of the Cartesian velocity with respect to relative coordinates Y . For example, if in Fig. 5.2, is shown in Fig. 7.1. q24 kq q =qY 1.7.3 RECURSIVE FORMULA FOR qTqG) (B g = Recursive formula for is obtained by using the recursive formula in Eq. 4-7. By replacing i by , Eq. 4-7 can be rewritten as q- (i 1) ) () (i iT1)i1 - (i 1 - ii iT1)i2 - (i 1)i - (iS G B SS G B g+ = + = (7-6) Taking partial derivative of Eq. 7-6 with respect to yields kq RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION (Y1)q24 = 0(B671)q24 = 0,(B672)q24 = 0(B461)q24 = 0,(B462)q24 = 0(B241)q24 ,(B242)q24 (B121)q24 = 0,(B122)q24 = 0(B231)q24 = 0,(B232)q24 = 0(B351)q24 = 0,(B352)q24 = 0(B281)q24 = 0,(B282)q24 = 0(Y2)q24 = 0(Y4)q24 =(B241)q24Y2+(B242)q24q24(Y3)q24 = 0(Y8)q24 = 0(Y5)q24 = 0 .(Y6)q24 = B461 (Y4)q24 (Y7)q24 = B671 (Y6)q24 Fig. 7.1 Computation Sequence for 24qY k k kk k k) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) (i iT1)i1 (i i iT1)i1 (i 1 ii iT1)i2 (i i iT1)i2 (i 1)i (iq q qq q qS G B S G B SS G B S G B g+ + + = + + + = (7-7) Since is a constant vector, . If .ncR G 0 Gq =k ( ) ( ) (k k kq IV q III q II U U ) i , and are not functions of . Therefore their partial derivatives with respect to are zero. As a result, Eq. 7-7 can be simplified to 1)i1 (iB1)i2 kq(iBkq k kk k) ( ) ( ) () ( ) ( ) (iT1)i1 (i 1 iiT1)i2 (i 1)i (iq qq qS B SS B g == (7-8) Since for the tree end bodies, by the second equation 0 Sq =k) (i0 Sq =k) (1 - i 1-17 of Eq. 7-8 for ( ) ( )k kq IV q II U i . Thus, for ( ) ( )k kq IV q II U i), Eq. 7-8 becomes gq k) (1)i (i1) (i + 0 Sq =k) (i( ) (( ) (T1)i1 (i 1 iT(i 1)i (ikkB SB gqq ==24 kq == 0= 04635(g35)q24 = 0(S3)q24 = 0(g23(S2)q24 = 0)q24 = 0(g12)q24 = (B122)( S(S12)q24 =(B121)(S2) q242) q242424 qq g G=0 = (7-9) If , body (ki q I belongs to set ( )kq II , and . Thus, Eq. 7-7 becomes ) ( )) ( )ii 1)i2kkSSqq (7-10) For example, if in Fig. 5.2, is shown in Fig. 7.2. q G B q24T 1(g67)q24 (S6)q24 28 (g28)q24 = 0(S2)q24 = 0(g24)q24 = (B242)q24 S4(S2)q24 =(B241)q24 S4(g46)q24 = 0(S4)q24 = 07Fig. 7.2 Computation Sequence for TB RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION 1.7.4 RECURSIVE FORMULAS FOR AND q qx) B ( X & & =v vx B X ) (& & = To obtain the recursive formula for and ( , Eq. 7-1 is partially differentiated with respect to and for qx) B (&kvx) B&nr ...,kq v 1, k = . 1)i (i 1)i2 (i 1 i 1)i1 (i 1 i 1)i1 (i1 i 1)i1 (i 1 i 1)i1 (i i) ( ) ( ) () ( ) ( ) ( + + + + =x B X B X BX B X B Xq q qq q qk k kk k k& & && & & (7-11) 1)i (i 1)i2 (i 1 i 1)i2 (i1 i 1)i1 (i 1 i 1)i1 (i i) ( ) () ( ) ( ) ( + + + =x B X BX B X B Xv vv v vk kk k k& && & & (7-12) The recursive formulas for and are obtained as in Appendix A by following the similar steps taken in the previous sections. qx) B (&vx) B (& 1.8. APPLICATION OF THE GENERALIZED RECURSIVE FORMULAS A library of the generalized recursive formulas is developed in Section 7. This section shows how the library can be utilized to compute the terms in and in Eqs. 6.9 and 6.12. Inspection of and reveals that the residual and partial derivatives of F , , , , , and need to be computed. Only and are presented in this section and the rest are omitted for simplicity of the presentation. HpHFHaFpH&q vFqq q& &FqF 1.8.1 COMPUTATION OF THE RESIDUAL F The generalized force , and the Cartesian acceleration need to be computed to obtain shown in Eq. 6-7. The term is obtained by applying the recursive formula in Eq. 7.1. The recursive formula with in Eq 4.7 can be applied to evaluate in since is a vector in Q ZTY&F)ncR .Y&G BTnrR (TQ Y M GZ + = &GF 1-19 1.8.2 COMPUTATION OF THE JACOBIAN qFIn Eq. 6-7, differentiation of matrix with respect to vector results in a three dimensional array. To avoid the complexity, Eq. 6-7 is differentiated with respect to a typical generalized coordinate . Thus, B qkq nr ....., 2, 1, k , ) () (kk kT TT T= + + + =q ZZ q qQ Y M BQ Y M B F&& (8-1) Since the term can be easily expressed in terms of the Cartesian coordinates, is obtained by applying the chain rule, as ) (TQ Z k)Tq ZQ ( kT T) ( ) (kB Q Q Z Z q Z = (8-2) where BqZ= is used and denotes the kth column of the matrix . The resulting equation for becomes kB BkqF nr ....., 2, 1, k ), ) ( () (kT TT Tkk k= + + + =B Q Y M BQ Y M B FZ Z qZ q q&& (8-3) The first term in Eq. 8-3 can be obtained by applying the recursive formula for , with , as explained in section 7.3. Collection of , for all k, constitutes ( , which is equivalent to . Matrix consists of columns which are vectors in . Therefore, the application of , where is each column of matrix , yields the numerical result of . Finally, the second term in Eq. 8-3 is also obtained by applying , where . G B qkTT( Z T T((Z B( M G=) (TQ Y M GZ + = &kTT) )ZT(Z T)ZQ ) )kTB Q Z Z ) B QZQ ncRT(Z(kYq +&B Q Z Z)TT)ZG BTQ ncT( BGTZT)ZQ G BT RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION 1.9. NUMERICAL EXAMPLES 1.9.1 A GOVERNOR MECHANISM The mechanism shown in Fig. 9.1 consists of seven bodies, a spring-damper, five revolute joints, and a translational joint. The material properties and spring and damping constants of the system are shown in Table 9-1. The mechanism has redundant constraints that are removed by the Gaussian elimination with full pivoting. Consequently, it has only 2 degrees of freedom. Dynamic analysis is carried out for 2 seconds with error tolerance of for the system. The Z acceleration of body 4 is drawn in Fig. 9.2. The result obtained by the other commercial program and that obtained by the proposed method are almost identical. The average step size, the numbers of residual function evaluations, and CPU time on SGI R3000 are shown in Table 9-2. The CPU time spent by the other commercial program is about 6 times larger than that by the proposed method. Note that the number of function evaluations of the proposed method is smaller than that of the other commercial program. 510 3 432561R4R2R3R5S1 S2R1T17YXZ0.160.50.20.10945o Fig. 9.1 A governor mechanism 1-21 Table 9-1 Inertia properties of the governor mechanism and spring and damping constants Mass xI yIzIxyIyzI zxI Body 1 (Ground) not necessary Body 2 200.0 25.0 50.0 25.0 0.0 0.0 0.0 Body 3 1.0 0.1 0.1 0.1 0.1 0.1 0.1 Body 4 1.0 0.1 0.1 0.1 0.1 0.1 0.1 Body 5 1.0 0.15 0.125 0.15 0.0 0.0 0.0 Body 6 0.1 0.1 0.1 0.1 0.0 0.0 0.0 Body 7 0.1 0.1 0.1 0.1 0.0 0.0 0.0 Spring constant 1000 Damping constant 30 Table 9-2 Integration output information Program TOL Average step size No. fevals CPU time (sec) Other 510 3 210 1 . 1 748 41 Proposed 510 3 210 2 . 1 441 7 PROPOSED OTHER Fig. 9.2 Z acceleration of Body 4 RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION 1.9.2 A MULTI-WHEELED VEHICLE A vehicle example shown in Fig. 9.3 is chosen to show the practicality of the proposed method. The vehicle runs on a bump whose radius is 0.3048(m). The system consists of a chassis and twelve road wheels and arms. The material properties and spring and damping constants are shown in Table 9-3. The road wheel and arm are considered as a single body. As a result, the system has 18 degrees of freedom. 11.1 m/sec0.3048m Fig. 9.3 A multi-wheeled vehicle Figure 9.4 shows the vertical acceleration of the chassis. It is shown that the proposed method and the other commercial program yield almost identical results. The average stepsize, number of residual function evaluations, and CPU time on SGI R3000 are shown in Table 9-4. It can be shown that the proposed method performs much smaller number of residual function evaluations with larger stepsizes, and the CPU time by the proposed method is much shorter than that by the other commercial program. Since there is no closed chain in the system, the governing equations of motion are formulated as an ODE problem by the proposed method. On the other hand, the equations of motion by the other commercial program are formulated as an DAE problem. The DAE problem is generally more difficult to solve than the ODE problem. This general argument is supported by the numbers of function evaluation and average stepsize. 1-23 Table 9-3 Inertial properties of the vehicle mechanism and spring and damping constants MassxI yI zI xyI yzI zxI Body 1 (Ground) not necessary Body 2 40773. 231800 60840 251700 - 234. -Body 3 ~ Body 14 340.27 32.86 20.76 26.85 0.0 0.0 0.0 Spring constant 200000 Damping constant 40000 Table 9-4 Integration output information Program TOL. Average step size No. fevals CPU time (sec) Other 410 310 4 1359 330 Proposed 410 310 6 . 6 1167 69 OTHER PROPOSED Fig. 9.4 Vertical acceleration of the chassis RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION 1.10. CONCLUSIONS The recursive formulas are generalized in this research. The velocity transformation method is employed to transform the equations of motion from the Cartesian to the joint spaces. Computational structure of the equations of motion is examined to classify all necessary computational operations into several categories. The generalized recursive formula for each category is then applied whenever such a category of computation is encountered. Since the velocity transformation method yields the equations of motion in a compact form and computational efficiency is achieved by the generalized recursive formulas, the proposed method is not only easy to implement but also efficient. A dynamic analysis algorithm using the backward difference formula (BDF) and the relative generalized coordinate is implemented using the library of generalized recursive formulas developed in this research. Numerical studies showed that obtained solutions were numerically stable and computation time was reduced by an order of magnitude compared to a well-known commercial program. 1-25 REFERENCES 1. J. Wittenburg, Dynamics of Systems of Rigid Bodies, B. G. Teubner, Stuttgart, 1977. 2. Hooker, W., and Margulies, G., The Dynamical Attitude Equtation for an n-body Satellite, Journal of the Astrnautical Science, Vol. 12, pp. 123-128, 1965. 3. R. Featherstone, The Calculation of Robot Dynamics Using Articulated-Body Inertias, Int. J. Roboics Res., Vol 2 : 13-30, 1983. 4. D. S. Bae and Edward J. Haug, A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems, Mech. Struct. and Machines, Vol. 15, No. 4, pp. 481-506 5. Potra, F. A. and Petzold, L. R., ODAE Methods for the Numerical Solution of Euler-Lagrange Equations. Applied Nume. Math., Vol. 10, pp. 397-413, 1992 6. Potra, F. A. and Rheinboldt, W. C., 1989, On the Numerical Solution of Euler-Lagrange Equations, NATO Advanced Research Workshop on Real-Time Integration Methods for Mechanical System Simulation, Snowbird, Utah, U. S. A.. 7. Jeng Yen, Edward J. Haug, and Florian A. Potra, 1990, Numerical Method for Constrained Equations of Motion in Mechanical Systems Dynamics, Technical Report R-92, Center for Simulation and Design Optimization, Department of Mechanical Engineering, and Department of Mathematics, The University of Iowa, Iowa City, Iowa. 8. Ming-Gong Lee and Edward J. Haug, 1992, Stability and Convergence for Difference Approximations of Differential-Algebraic Equations of Mechanical System Dynamics, Technical Report R-157, Center for Simulation and Design Optimization, Department of Mechanical Engineering, and Department of Mathematics, The University of Iowa, Iowa City, Iowa. 9. Lin, T. C. and Yae, K. H., 1990, Recursive Linearization of Multibody Dynamics and Application to Control Design, Technical Report R-75, Center for Simulation and Design Optimization, Department of Mechanical Engineering, and Department of Mathematics, The University of Iowa, Iowa City, Iowa. RecurDyn/SOLVER THEORETICAL MANUALGENERALIZED RECURSIVE FORMULATION APPENDIX A : RECURSIVE FORMULAS Recursive formulas ) ( ikq I ) ( ikq II q qBx X ) ( = 1)i (i 1)2 (i1 i 1)i1 (i ikk k) () ( ) ( +=x BX B Xqq q k k1 i 1)i1 (i i q q) (X B ) (X = qTqG) (B g = iTi i iiTi i i ikk( S B SS B gq qq q) ( )) ( ) (1 ) 1 ( 12 ) 1 ( ) 1 (kk == 0 S0 gqq==kk) () (1 i1)i (i q qx) B ( X & & = 1)i (i 1)i2 (i1 i 1)i1 (i1 i 1)i1 (i ikkk k) () () ( ) ( ++=x BX BX B Xqqq q&&& & kk k) () ( ) (1 i 1)i1 (i1 i 1)i1 (i iqq qX BX B X +=&& & v vx B X ) (& & = 1)i (i 1)i2 (i1 i 1)i1 (i ikk k) () ( ) ( +=x BX B Xvv v&& & kk k) () ( ) (1 i 1)i2 (i1 i 1)i1 (i ivv vX BX B X +=&& & Recursive formulas ) ( ikq III ) ( ikq IV q qBx X ) ( = 0 Xq =k) (i 0 Xq =k) (i q qG B g ) (T= k kk k) ( ) () ( ) (iT1)i1 (i 1 iiT1)i2 (i 1)i (iq qq qS B SS B g == 0 S0 gqq==kk) () (1 i1)i (i q qx B X ) (& & = 0 Xq =k) (i& 0 Xq =k) (i& v vx B X ) (& & = 0 Xv =k) (i& 0 Xv =k) (i& Recursive formulas ) ( ikq I or i or or ) (kq II ) ( ikq III ) ( ikq IV Bx X = 1)i (i 1)i2 (i 1 i 1)i1 (i i + = x B X B X G B gT= ) (0) (1 i 1 iT1 ) 1 i ( i in1 i 1 iT2 ) 1 i ( i ) 1 i ( i+ + ++ + + ++ == + =S G B SSS G B g x B X & & = i ) 1 i ( 2 i ) 1 i ( 1 i 1 i ) 1 i ( 1 i 1 i ) 1 i ( i + + = x B X B X B X & & & & 1-27 RecurDyn/SOLVER THEORETICAL MANUAL GENERALIZED RECURSIVE FORMULATION 2 DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION 2.1. INTRODUCTION The dynamic behavior of a constrained mechanical system is often represented by differential algebraic equations (DAEs)[1]. Solutions of DAEs are generally more difficult to obtain than those of ordinary differential equations (ODEs)[2]. To solve DAEs, a direct discretization method was proposed by Gear[3]. Since the solution obtained by Gear does not satisfy the velocity level constraints, consistent initial conditions cannot be obtained. It was found that the inconsistency often resulted in a poor local error estimation[4]. A series of stabilization methods[5-7] which employ either Lagrange multipliers or constraint violation penalty terms were followed. Recently several solution methods[8], projecting the differential equations on the inflated constraint manifolds, have appeared. Two kinds of solution process are available. In the first solution process, the numerical integration is carried out first and the integrated variables are corrected so that the position level constraints, the velocity level constraints, and the acceleration level constraints are satisfied. Since the correction is made sequentially level-by-level, the size of system equations to be solved remains small. However, the integration stepsize can be excessively small for highly nonlinear or stiff problems due to a narrow stability region of the explicit method. In order to overcome this difficulty, the second solution process is developed. In the second solution process, the numerical integration formula, kinematic constraints and their derivatives, and equations of motion are solved simultaneously. Therefore, the size of the system equations to be solved becomes larger although the problem of excessive small step size is resolved. In addition to the problem of large size of the matrix equation, the condition of the matrix becomes poor as the stepsize gets smaller for discontinuous systems. The poor condition of the matrix often RecurDyn/SOLVER THEORETICAL MANUALDECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION results in large error in the solution of the matrix equation. In this paper, a decoupling solution method for the implicit numerical integration method is proposed. This method is free from the problems of the poor matrix condition and the excessively small step size as well as the large matrix size. In section 2, overdetermined DAEs for constrained mechanical systems are given. A decoupling solution method is given in section 3. In section 4, the numerical algorithm is provided. The numerical examples are given in section 5 to demonstrate the efficiency of the proposed method. Conclusions are drawn in section 6. 2.2. IMPLICIT NUMERICAL INTEGRATION FOR DIFFERENTIAL ALGEBRAIC EQUATIONS The equations of motion for a constrained mechanical system can be implicitly described as 0 q v = (2.1.a) 0 ) a, v, F(q, = (2.1.b) 0 (q) = (2.1.c) where is the generalized coordinate vector in Euclidean space qnR , and is the Lagrange multiplier vector for constraints in mR , represents the position level constraint vector in mR , and its Jacobian is expressed that is assumed to have full row-rank. Successive differentiations of Eq. 2.1.c yield velocity and acceleration level constraints, nq mR 0 v v) (q,q = = (2.2.a) 0 a a) v, (q, = =q & (2.2.b) Equations 2.1 and 2.2 comprise a system of overdetermined differential algebraic 2-3 equations (ODAE). An algorithm based on backward differentiation formula (BDF) to solve the ODAE is given in Ref. 1 as follows: 0 q v U v a U(q) v a ) a, v, F(q,R UR UF(x)H(x)T2T1qq2T21T1=((((((((((((

||.|

\| ||.|

\| =((((((((((((

||.|

\|||.|

\|=201000bhbhbhbh&& & (2.3) where = =k1 i1 n i01bb1v , = =k1 i1 n i02bb1q [ x =iUTq, k is the order of integration and are the BDF coefficients. Here, and the columns of constitute bases for the parameter space of the position and velocity level constraints. are chosen so that has an inverse. Therefore, the parameter space spanned by the columns of and the subspace spanned by the columns of constitute the entire space ibiU] q , v , a ,T T T T

) 2 , 1 i () m n ( n= R(((TqUiiUnR . The number of equations and the number of unknowns in Eq. 2.3 are the same, so Eq. 2.3 can be solved. Newton's numerical method can be applied to obtain the solution . x i i iH x Hx = (2.4.a) i i 1 ix x x + =+ (2.4.b) LU-decomposition of the matrix not only increases the computation time ixH RecurDyn/SOLVER THEORETICAL MANUALDECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION but also produces an ill-conditioned matrix as h approaches zero [4]. In order to eliminate these problems, Eq. 2.4.a will be divided into several pieces to obtain , , and q v a separately in the next section. h aFa mR1 v 2.3. A DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION Equation 2.4.a can be rewritten in detail as follows: 0 x F F a F v F q F a v q = + + + + ) ( (3.1.a) 0 x q v a q v q = + + + ) (& & & & & & (3.1.b) 0 x q q = + ) ( (3.1.c) 0 x q v q q = + + ) (& & (3.1.d) 0 x R v a U = + )) ( (1T1h h (3.1.e) 0 x R q v U = + )) ( (2 2h hT (3.1.f) where 0bh1U. Equation 3.1.e can be rewritten in an equivalent inflated form by choosing such that . as follows [7]: 0Tq1aT1 = F U 0 F U x R v = + + 111) (Tq aTih h h (3.2) where is a mass matrix and is generally nonsingular. The can be singular if a parametric formulation is employed. If is singular, Eqs. 3.1 must be solved simultaneously to obtain aFaFq , v , a and . The vector is a new unknown variable. The a is thus obtained from Eq. 3.2 in terms of as 1T 1a 1) (h1 F x R v aq = (3.3) 2-5 Substituting Eq. 3.3 into Eq. 3.1.a yields ) ( ) ( ) (h1x R F x F vFF q Fa 1Tqav q + = + |.|

\|+ + (3.4) Equation 3.1.f can be rewritten in an equivalent inflated form by choosing such that 2U0 FF UTqav = |.|

\|+ 1T2h as follows: 0 FF x R q vTqav 2 = |.|

\|+ + + 21) (hh h h (3.5) where h+avFF is assumed to be a nonsingular matrix and is a new unknown variable. The solution process for the case of a singular matrix will be explained later in this section. Equation 3.5 can be solved for in terms of as follows: m2R v q 212h) (h1 FF x R q vTqav|.|

\|+ = (3.6) Substituting Eq. 3.6 into Eq. 3.4 and multiplying both sides of Eq. 3.4 by yields 2h32h R q KTq*= + (3.7) where a v q*F F F K + + h h2 (3.8.a) 2 1 (3.8.b) ) ( ) h ( h ) ( h ) ( h2 12 23x R F F x R F x F Ra v a + + + (3.8.c) RecurDyn/SOLVER THEORETICAL MANUALDECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION Equations 3.7 and 3.1.d are combined to obtain ((

=((

(((

) ( h 032x Rq KqTq* (3.9) Equation 3.9 is then solved for and . Note that is scaled by to make the coefficient matrix of Eq. 3.9 ill-conditioned, even as approaches to zero. q 2hhMultiplying both sides of Eq. 3.5 by h+avFF yields ) (h h1h2 2x R qFF vFFavTqav |.|

\|+= + |.|

\|+ (3.10) Equations 3.10 and 3.1.c are combined to obtain ((((

|.|

\|+ =((

(((

+q x x R qFFv FFqavqTqav) () (h h10h22 (3.11) where has been obtained from Eq. 3.9. Equation 3.11 is solved for the and . Multiplying both sides of Eq. 3.3 by yields q 2v aF |.|

\| = + v x R F a Fa ah1) (1 1Tq (3.12) Equations 3.12 and 3.1.b are combined to obtain 2-7 ((((

|.|

\| =((

(((q v x v x R Fa Fq vaqTq a& & & & & &) (h1) (011

+ + =iyixiziyixizixiziyixizixiyixiziyixizixiziyixizixiyiziyiziyif cos cos sin sin cos sin sin cos sin cos sin sincos sin sin sin sin cos cos cos sin sin sin cossin sin cos cos cosA (4) If is infinitesimal, the matrix can be approximated as |Tiziyixi = |ifA ((((

111ixiyixiziyizif A (5) The rotational deformation vector can be represented by linear combination of rotational mode shapes of body as ii ifi ip = (6) where is a modal matrix whose columns are composed of rotational mode i 3-7 shapes and is the vector of modal coordinate. Ciq |

=000ifpFinally, kinematic constraints between two body frames of the flexible and virtual bodies can be obtained from Eqs. (1) and (3) as follows: ( ) 0 u u A R r C = + = + ifi i i i iR 01 (7) 0h A A f h A A fh A A g h A A gg A A f g A A f=(((((

=+ ++ ++ +1 , 11 , 11 , 1i i ifT iTi Ti i ifT iTi Ti i ifT iTi T (8) where | |((((

=1 0 00 1 00 0 1h g f (9) Orthogonality conditions would have been used in deriving the orientational constraints. However, the in Eq. (8) is employed in this research for simple implementation. Eqs. (7) and (8) yields algebraic constraint equations that describe the flexible joint between flexible body and virtual body . Taking variation of Eqs. (7) and (8) yields iCi 1 + i ( ) ( )( )0 qCCq Cqqq =(((

= iiR iflexi (10) where | |TTiTiTifTiTi i 1 1 + += r p R (11) and the constraint Jacobian matrix ( )flexiqC is obtained as ( ) |( )(((( = +++111ihT i fhT ihTihT i fhT ihTigT i fgT igTiiRi i iRB f 0 B f B fB g 0 B g B gB f 0 B f B fC0 I A B I Cqq (12) RecurDyn/SOLVER THEORETICAL MANUALFLEXIBLE MULTIBODY DYNAMICS USING A VIRTUAL BODY AND JOINT where h g kskewskewskewskewiTi iki i fki iTi iki i i, ,) () () () (1 11 ,1=)`====+ +++k A A Bk A BA k A A Bu A B (13) and the vectors ) (iskew u , , , and are the skew symmetric matrices of vectors, ) (1k A + iskew ) (1 ,k A + i iskew ) (k skewiu , , , and , respectively. In order to obtain the acceleration level constraint, one can differentiate Eqs. (7) and (8) twice with respect to time to yield k A1 + iA1 , + i ik k ( ) ( ) ( ) ( )( ) ( )((((((

+ + + + + + + + + == =4 3 2 14 3 2 14 3 2 1222) ( 2 ) (hThThThThThThThTgTgTgTgTifiRi i iflexici iflexi iflexiskew skewH f H f H f H fH g H g H g H gH f H f H f H f p A u AQ q q C q Cqq q&& & & & (14) where the is the angular velocity with respect to the body reference frame and the generalized velocity vector is q& | |TTiTiTifTiTi i 1 1 + += r p R q & &&& (15) and ( ) ( )h g kskewskew skewskew skew skewskew skewifi i i ififiki i iTiki iTi ikiTi i ik, ,) () ( ) () ( ) ( ) () ( ) (1 , 41 1 1 31 1 21 1=)` ====++ + ++ ++p k A A p Hk A A H k A A Hk A A H& & (16) 3-9 3.3. EQUATIONS OF MOTION Even though the proposed method is applicable to a general system consisting of many flexible bodies, a slider crank mechanism with one flexible body in Fig. 4(a) is used to clearly show the impact of the proposed method on the equations of motion. An equivalent virtual system, modeled by using the rigid virtual bodies proposed in this investigation, is shown in Fig. 4(b). The augmented equations of motion for the system is obtained by using the general form of equations of motion as [11] ((

+ +=((

(((

cs v eTQQ Q Qq0 CC Mqq & & (16) where is the mass matrix of the system. The vector consists of translational acceleration for rigid and flexible bodies, angular acceleration, and modal acceleration for the flexible body. M q& & FLEXIBLE BODY FLEXIBLE BODYRIGID BODY RIGID BODYCRANKCRANKSLIDER SLIDERCOUPLERCOUPLER123YZXP1 P1 (a) Two rigid bodies and one flexible body RecurDyn/SOLVER THEORETICAL MANUALFLEXIBLE MULTIBODY DYNAMICS USING A VIRTUAL BODY AND JOINT FLEXIBLE BODY FLEXIBLE BODYRIGID BODY RIGID BODYVIRTUAL BODY VIRTUAL BODYCRANKCRANKCOUPLERCOUPLERSLIDER SLIDERVIRTUAL BODY VIRTUAL BODY 12453 (b) Two rigid bodies, one flexible body and two virtual bodies Figure 4 Slider crank mechanism with one flexible body The is the vector of Lagrange multipliers and , , , and are the strain energy terms, velocity induced forces, externally applied forces, and the vector absorbs terns that are quadratic in the velocities, defined clearly by Shabana [11]. sQvQeQcQcQ 3.3.1 COEFFICIENT MATRIX OF CONVENTIONAL AUGMENTED FORMULATION The mass matrix for the system in Fig. 4(b) is ((((

=321rrfM 0M0 MM (17) where the mass matrices for virtual bodies, and are the mass matrix for flexible body and for a rigid body, as fMrM 3-11 ) 3 , 2 ( ,6 6) 6 ( ) 6 (1=((

=((((

=+ +ksymmetrickkrr krnf nfff f frrrrf m 00 mm m mm mmMM (18) where is the number of modal coordinates. The constraint Jacobian matrix of the slider crank mechanism with flexible crank is nfC) (qC ( )( )( )( ) ((((((

=int30int23,12,01) (jojoc flexc flexcqqqqqCCCCC (19) where, ( )c flex,qC is the constraint Jacobian matrix of the flexible joint obtained by the conventional method[11]. 3.3.2 COEFFICIENT MATRIX OF PROPOSED AUGMENTED FORMULATION The mass matrix for the system in Figure 4b is (((((((

=54321rrvfvMM 0M0 MMM (20) where the mass matrix for virtual body, , the mass matrix for flexible body, vM RecurDyn/SOLVER THEORETICAL MANUALFLEXIBLE MULTIBODY DYNAMICS USING A VIRTUAL BODY AND JOINT fM , and the mass matrix for rigid body, are rMsymmetricm,6 6(((((((((intintint,,intpp | | 3 , 1 ,6 6 = = kv0 M ) 6 ( ) 6 (2nf nfff f frrrrf+ +((((

=m mm mmM (21) ) 5 , 4 ( =((

= kkkrr krm 00 mM The proposed constraint Jacobian matrix ( )pqC of the slider crank mechanism with flexible crank is ( )( )( )( )( )( )( )

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